We are given a quadratic function in the form:

$y = a(x - b)^2 + c$

with the following information:

1. The function passes through the points $(-2, 0)$ and $(6, 0)$.
2. The maximum value of $y$ is 48.

Step 1: Use the fact that the function passes through the points

Since the quadratic passes through the points $(-2, 0)$ and $(6, 0)$, we know these are the roots of the quadratic. Therefore, the vertex of the parabola lies halfway between these two points.

The midpoint between $(-2, 0)$ and $(6, 0)$ is:

$b = \frac{-2 + 6}{2} = 2$

Thus, the value of $b$ is $2$, which represents the $x$-coordinate of the vertex.

Step 2: Use the maximum value of $y$

The maximum value of the quadratic is given as 48. Since the vertex occurs at $x = 2$, substitute $x = 2$ and $y = 48$ into the equation $y = a(x - b)^2 + c$:

$48 = a(2 - 2)^2 + c$ $48 = 0 + c$ $c = 48$

Thus, $c = 48$.

Step 3: Use the fact that the points $(-2, 0)$ and $(6, 0)$ are on the curve

We now know that the quadratic passes through the points $(-2, 0)$ and $(6, 0)$, so we can substitute one of these points into the equation to find $a$. Let's use the point $(-2, 0)$:

Substitute $x = -2$ and $y = 0$ into the equation:

$0 = a(-2 - 2)^2 + 48$ $0 = a(-4)^2 + 48$ $0 = 16a + 48$ $16a = -48$ $a = -3$

Final Answer:

The values of $a$, $b$, and $c$ are:

$a = -3, \quad b = 2, \quad c = 48$

Would you like a more detailed explanation of any step, or do you have any further questions?

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5 Related Questions:

1. How does the value of $a$ affect the orientation of the parabola?
2. What happens to the quadratic if the vertex value $y = 48$ were increased or decreased?
3. How can you write the equation of a quadratic in standard form from its vertex form?
4. How do you determine the axis of symmetry of a quadratic function?
5. How can you find the roots of a quadratic equation if the vertex and maximum value are known?

Tip:

When a quadratic function is in vertex form, $y = a(x - b)^2 + c$, the vertex is located at $(b, c)$, and the axis of symmetry is the vertical line $x = b$.